Theorem REF  follows from a technical result we present later,
i.e., Theorem REF , which we use to establish spectral independence
for the zero external field Ising model \(\mu _G\)  with \(\beta \in {\mathbb {M}_{\rm Ising}}({\rho }_G,\delta )\) .
Once we establish spectral independence for \(\mu _G\) , we use results from [1]} to derive
the bound on the mixing time.
